We now have all the difficult things out of the way. From here, it is easier and more relevant to the main question that statistics answers in Psychology.
We go back to the equation from before:
and this time we will give a meaning to it. Each i is a participant; yi and xi are measurements of variables from that participant. We are using x to try and explain y.
This equation leads to 3 variances: var(y), var(b.x) and var(e), that are related by:
and var(b.x) is the part of var(y) explained by x and var(e) is the remaining unexplained variance. We want to know how good x is at explaining y: what proportion of the variance of y is explained by the variance of x.
We can see a simple way of doing this:
This looks easy, but we don’t have a value for b.
In the previous page, we showed that:
and from this, we have:
and
leading to: